Moisture Absorption in Polymer Composites Reinforced with Vegetable Fiber: A Three-Dimensional Investigation via Langmuir Model

This work aims to study numerically the moisture absorption in polymer composite reinforced with vegetable fibers using the Langmuir model which considers the existence of free and entrapped water molecules inside the material. A three-dimensional and transient modeling for describing the water absorption process inside the composite and its numerical solution via finite volume method were presented and discussed. Application has been made for polymer composites reinforced with sisal fiber. Emphasis was given to the effect of the layer thickness of fluid close to the wall of the composite in the progress of water migration. Results of the free and entrapped solute (water) concentration, local moisture content and average moisture content, at different times of process, and inside the composite were presented and analyzed. It was verified that concentration gradients of the molecules (free and entrapped) are higher in the material surface, at any time of the process, and concentration of free solute is greater than the concentration of entrapped solute. It was verified that the water layer thickness surrounding the composite strongly affects the moisture absorption rate.


Introduction
Composite is a material consisting of two or more insoluble materials, which are combined to create a useful engineering material having some properties not obtained by the constituents taken separately [1].
The main purpose to manufacture composite materials is to obtain a new material with increased properties, mainly from the mechanical point of view (mechanical resistance) and low weight [2].
Among the different types of composite materials, we can cite those which are reinforced with particles, reinforced with fibers (high length/diameter ratio) and those structural (combination of composites and homogeneous materials whose properties depend on the geometric designs of the structural elements). Therefore, the objective of this research is to predict mass transfer during the process of water absorption in polymer composites (parallelepipedic shape) reinforced with vegetable fibers, placed randomly into the matrix by using the Langmuir model, including the effect of water layer thickness at the surface of the solid.

The Physical Problem and Geometry
The physical problem to be studied is the absorption of water into a porous solid material, with a geometry of a parallelepiped with dimensions 2R x × 2R y × 2R z , immersed in a fluid medium, distant from the container wall by l x (in x-direction), 1 y (in y-direction), and l z (in z-direction), as shown in Figure 1. Therefore, the objective of this research is to predict mass transfer during the process of water absorption in polymer composites (parallelepipedic shape) reinforced with vegetable fibers, placed randomly into the matrix by using the Langmuir model, including the effect of water layer thickness at the surface of the solid.

The Physical Problem and Geometry
The physical problem to be studied is the absorption of water into a porous solid material, with a geometry of a parallelepiped with dimensions 2Rx × 2Ry × 2Rz, immersed in a fluid medium, distant from the container wall by lx (in x-direction), 1y (in y-direction), and lz (in z-direction), as shown in Figure 1.

Mathematical Model
For the analysis and solution of the physical problem studied in this paper, the following considerations were made: a) the solid is homogeneous and isotropic, b) it has a parallelepipedic shape, c) the mass diffusion coefficient remains constant throughout the process, d) the process is transient, e) variations in material dimensions during the water absorption process were neglected , f) mechanism of water transport inside the material is purely diffusive, g) there is no mass generation, and h) the solid is completely dry at the beginning of the process. It is noteworthy that the development of the study considered the composite homogeneous, isotropic, and without direct influence of fiber orientation within the composite. This was due to the manufacturing process of the sisal fiber-reinforced polymeric composite developed by Santos [37], used in the experimental phase to validate this research: hand lay-up (randomly distributed fibers). Thus, to contemplate the properties of the fibers in the numerical solution, their characteristic was coupled to the mass diffusion coefficient.
In the Langmuir model, the non-Fickian moisture absorption behavior can be explained quantitatively by assuming that moisture absorption occurs in the presence of two simultaneous stages, one being the free water stage and the another the entrapped water stage [26], described by the following mass transfer Equations: In Equations (1) and (2), C represents the concentration of the free solute to diffuse into the material, S represents the concentration of the entrapped solute, D c is the mass diffusion coefficient

Mathematical Model
For the analysis and solution of the physical problem studied in this paper, the following considerations were made: a) the solid is homogeneous and isotropic, b) it has a parallelepipedic shape, c) the mass diffusion coefficient remains constant throughout the process, d) the process is transient, e) variations in material dimensions during the water absorption process were neglected, f) mechanism of water transport inside the material is purely diffusive, g) there is no mass generation, and h) the solid is completely dry at the beginning of the process. It is noteworthy that the development of the study considered the composite homogeneous, isotropic, and without direct influence of fiber orientation within the composite. This was due to the manufacturing process of the sisal fiber-reinforced polymeric composite developed by Santos [37], used in the experimental phase to validate this research: hand lay-up (randomly distributed fibers). Thus, to contemplate the properties of the fibers in the numerical solution, their characteristic was coupled to the mass diffusion coefficient.
In the Langmuir model, the non-Fickian moisture absorption behavior can be explained quantitatively by assuming that moisture absorption occurs in the presence of two simultaneous stages, one being the free water stage and the another the entrapped water stage [26], described by the following mass transfer Equations: In Equations (1) and (2), C represents the concentration of the free solute to diffuse into the material, S represents the concentration of the entrapped solute, D c is the mass diffusion coefficient (free molecules), t is the time, λ is the probability of a free solute molecule to be entrapped inside the solid, and µ is the probability that an entrapped solute molecule becomes free.
For the solution of Equations (3) and (4), the following initial and boundary conditions were used: Initial conditions: it was considered that the solid is completely dry at the beginning of the process. Thus, one can write: Boundary conditions: on the surface of the solid, the variation in the concentration of the solute in the fluid medium is considered to be equal to the diffusive flux of solute at the surface of the material. Thus, one can write: Once determined C and S at any point inside the material, it is possible to calculate the total moisture content present in the material at any position and instant of time. Thus, the total moisture content is given by the sum of the concentrations of C and S, as follows: Furthermore, the average moisture content of the solid at any time of the process is given by: (10) in which V is the total volume of the solid and dV = dxdydz is the volume of an infinitesimal sample of the porous solid.

Numerical Solution
Basically, the numerical solution of a partial differential equation consists of two steps: a) discretizing the physical domain under study in several sub domains and b) transforming the governing equation into a linear algebraic equation in the discretized form applied to each sub domain contained in the solid under study. After these procedures, the result is a set of linear algebraic equations whose solution provides the distribution of the variable of interest within the domain and in time.
For the three-dimensional numerical solution of the governing equations applied to the absorption of moisture in materials with parallelepipedic form, the finite volume method was used. In this solution, a fully implicit formulation for the concentration of free solute and explicit formulation for the concentration of entrapped solute were used. For this, due to symmetrical equivalence, only 1/8 of the solid was used. From this symmetric equivalence and the parallelepiped geometry of the studied composite, the numerical mesh was developed to 1/8 of the solid, as shown in Figure 2, with the refinement and choice of mesh to be determined later. Thus, for each control volume, a solution was developed for the properties of the mass phenomena studied from the Langmuir model.  Figure 3 illustrates a control volume (sub domain) used for the discretization of governing equations. Further, Figure 3 shows the nodal point P (in the center of the control volume), its adjacent neighbors W, E, S, N, T, and F, the distances between these nodal points, and the dimensions Δx, Δy, and Δz, of the control volume. Following is the detailing of the whole numerical procedure used for the solution of the governing equations.
Solution for the water concentration.
The numerical solution of Equation (3) is obtained by integrating it in volume and time, as follows: Assuming a fully implicit formulation, one can write the result of Equation (11) in the discretized form as follows:  Figure 3 illustrates a control volume (sub domain) used for the discretization of governing equations. Further, Figure 3 shows the nodal point P (in the center of the control volume), its adjacent neighbors W, E, S, N, T, and F, the distances between these nodal points, and the dimensions ∆x, ∆y, and ∆z, of the control volume.  Figure 3 illustrates a control volume (sub domain) used for the discretization of governing equations. Further, Figure 3 shows the nodal point P (in the center of the control volume), its adjacent neighbors W, E, S, N, T, and F, the distances between these nodal points, and the dimensions Δx, Δy, and Δz, of the control volume. Following is the detailing of the whole numerical procedure used for the solution of the governing equations.
Solution for the water concentration.
The numerical solution of Equation (3) is obtained by integrating it in volume and time, as follows: Assuming a fully implicit formulation, one can write the result of Equation (11) in the discretized form as follows: Following is the detailing of the whole numerical procedure used for the solution of the governing equations.
Solution for the water concentration.
The numerical solution of Equation (3) is obtained by integrating it in volume and time, as follows: x y z t Assuming a fully implicit formulation, one can write the result of Equation (11) in the discretized form as follows: where It should be noted that Equation (12) is only applied to the internal control volumes of the computational domain ( Figure 4). For the other control volumes (symmetry and border), it is proceeded with a mass balance in each one of them. In total, there are 27 different types of control volume of dimensions ∆x, ∆y, and ∆z. As an example, the result of Equation (11) applied to the control volume of the right upper corner of the computational domain, as shown in Figure 5, is given by: where It should be noted that Equation (12) is only applied to the internal control volumes of the computational domain ( Figure 4). For the other control volumes (symmetry and border), it is proceeded with a mass balance in each one of them. In total, there are 27 different types of control volume of dimensions x, y Δ Δ , and z. Δ As an example, the result of Equation (11) applied to the control volume of the right upper corner of the computational domain, as shown in Figure 5, is given by: It should be noted that Equation (12) is only applied to the internal control volumes of the computational domain ( Figure 4). For the other control volumes (symmetry and border), it is proceeded with a mass balance in each one of them. In total, there are 27 different types of control volume of dimensions x, y Δ Δ , and z. Δ As an example, the result of Equation (11) applied to the control volume of the right upper corner of the computational domain, as shown in Figure 5, is given by:  Bulleted lists look like this: Solution for the entrapped water concentration. The numerical solution of Equation (4) is obtained by integrating it in volume and time, as follows: Assuming an explicit formulation, one can write the Equation (29), in the discretized form, as follows: where In the discretized form, the local and average moisture contents can be written, respectively, as follows:  (35) in which i, j and k represent the position of the nodal point in the x, y, and z directions, respectively, and npx, npy, and npz are the nodal point numbers in the x, y, and z directions, respectively. From the discretization of the governing equations, a system of algebraic equations is generated that must be solved to obtain the values of C, S, and M within the material throughout the process. The solution of this system of algebraic equations was done using the Gauss-Seidel iterative method. In order to obtain the numerical results, a computer code was developed in Mathematica software (version 9).

Mesh and Time Step Refinements
For the refinement study of the mesh and time step, isothermal cases were used, with µ = 3.27972 × 10 −6 s −1 , λ = 2.12852 × 10 −6 s −1 , and D = 7.31787 × 10 −12 m 2 /s. A total time of approximately 56 h was considered for analysis. The values of µ, λ, and D were initially estimated using the methodology proposed by Joliff et al. [16], when applied to water absorption in composite materials reinforced by sisal fiber [30,31].
Briefly  Table 1 summarizes all the physical parameters of the composite and water used in the mesh refining simulations.  Table 2 summarizes all the physical and geometric parameters of the composite and water used in the refining simulation of the time step.

Validation
For the validation of the mathematical modeling and numerical solution of the governing equations, a comparison was made between the numerical results of the average moisture content obtained in this paper and the analytical (Fick's model, three-dimensional) and experimental data reported by Santos [37], for the water absorption process in polymer composites reinforced by sisal fibers. For preparation of the samples, a polymer material was used, an unsaturated polyester resin (Resapol 10-316), with low viscosity, and pre-accelerated. This resin is reticulated by styrene peroxide, using as initiator Methyl Ethyl Ketone (MEK-P) at a concentration of 1% by weight. The method used was hand lay-up, in which the fibers were arranged randomly and the molding of the composite was performed by compression.
To obtain the Fickian model from the Langmuir model, the ∂S ∂t term, in Equation (1), was made null and, in addition, high values of the sample distance to the container wall (l x , l y , and l z ), in Equation (6), were used. The objective was to reduce the effect of these parameter on the obtained results, since this effect is not considered in the Fick's model. It has been found that this consideration allows that the concentration of the free solute in the water does not change over time, being constant and equal to the equilibrium concentration ∂S ∂t 0. Equation (36) represents the three-dimensional (3D) analytical solution for the average moisture content of a parallelepipedic solid with equilibrium boundary condition, as reported by Santos [37].
in which M ∞ is the equilibrium moisture content, M o is the initial moisture content of the material, and B x , B y , and B z are coefficients given as follows: parameters β m , β n , and β k in Equation (34) are the calculated eigenvalues respecting the equilibrium boundary conditions at the surface of the solid, as follows: then, by using Equation (38)-(40), we obtain: Table 3 shows the data used in the simulation for validation with the Fickian model. An isothermal case was used, µ = λ = 0, D = 3.04 × 10 −12 m 2 /s and M ∞ = 0.1468193 kg/kg, as reported by Santos [37]. Table 3. Parameters used in the simulation for the three-dimensional (3D) case.

Case
Mesh

Arbitrary Cases
In this item, the influence of the geometrical parameters l x , l y , and l z (see Figure 1) in the process of water absorption in polymeric composites reinforced by fiber was evaluated. This is an unprecedented analysis, being the great contribution of this paper. For this analysis, it was considered µ = λ = 1 × 10 −6 s −1 and D = 1 × 10 −12 m 2 /s. Table 4 summarizes the process parameters used in the simulations. Table 4. Parameters used in the simulation of arbitrary cases.

Case
Mesh

Study of Mesh and Time
Step After analyzing these figures, it was observed that, for the three meshes used in the prediction of water absorption, there is no significant variation in the predicted results with the greater refinement of the mesh. Thus, considering that the greater the refinement of the mesh, the longer the computational time to reach the conclusion of the simulation, the mesh with the smallest number of nodal points was chosen, without loss of information on the behavior of the process parameters. Similar behavior was found with the time step study (Figure 9), where ∆t = 20 s was chosen for the next analyzes.

Study of Mesh and Time
Step     After analyzing these figures, it was observed that, for the three meshes used in the prediction of water absorption, there is no significant variation in the predicted results with the greater refinement of the mesh. Thus, considering that the greater the refinement of the mesh, the longer the computational time to reach the conclusion of the simulation, the mesh with the smallest number of  After analyzing these figures, it was observed that, for the three meshes used in the prediction of water absorption, there is no significant variation in the predicted results with the greater refinement of the mesh. Thus, considering that the greater the refinement of the mesh, the longer the computational time to reach the conclusion of the simulation, the mesh with the smallest number of

Validation: Application of the Langmuir Model as Fick's Model
The numerical results strongly depend on the boundary conditions, thermophysical properties, and geometry considered. Figure 10 illustrates the comparison between the results of the average moisture content obtained with the numerical solution, from the approximation of the Langmuir model to the Fick's model, and experimental and analytical data (Fick's model) reported by Santos [37]. After analyzing Figure 10, it was observed an excellent agreement between the data predicted by the numerical solution, with those analytical and experimental, reported by Santos [37], which shows that the mathematical model presents an adequate description of the diffusion process inside the material. Therefore, the Fick's model is a particular case of the Langmuir diffusion model.

Application to Arbitrary Cases
The absorption of water is facilitated when the polymer molecules have groups capable of forming hydrogen bonds. Vegetable fibers are rich in cellulose, hemicellulose, and lignin which have hydroxy groups, thus have high affinity for water. The absorption of water by the resin, in turn, can

Validation: Application of the Langmuir Model as Fick's Model
The numerical results strongly depend on the boundary conditions, thermophysical properties, and geometry considered. Figure 10 illustrates the comparison between the results of the average moisture content obtained with the numerical solution, from the approximation of the Langmuir model to the Fick's model, and experimental and analytical data (Fick's model) reported by Santos [37].

Validation: Application of the Langmuir Model as Fick's Model
The numerical results strongly depend on the boundary conditions, thermophysical properties, and geometry considered. Figure 10 illustrates the comparison between the results of the average moisture content obtained with the numerical solution, from the approximation of the Langmuir model to the Fick's model, and experimental and analytical data (Fick's model) reported by Santos [37]. After analyzing Figure 10, it was observed an excellent agreement between the data predicted by the numerical solution, with those analytical and experimental, reported by Santos [37], which shows that the mathematical model presents an adequate description of the diffusion process inside the material. Therefore, the Fick's model is a particular case of the Langmuir diffusion model.

Application to Arbitrary Cases
The absorption of water is facilitated when the polymer molecules have groups capable of forming hydrogen bonds. Vegetable fibers are rich in cellulose, hemicellulose, and lignin which have hydroxy groups, thus have high affinity for water. The absorption of water by the resin, in turn, can After analyzing Figure 10, it was observed an excellent agreement between the data predicted by the numerical solution, with those analytical and experimental, reported by Santos [37], which shows that the mathematical model presents an adequate description of the diffusion process inside the material. Therefore, the Fick's model is a particular case of the Langmuir diffusion model.

Application to Arbitrary Cases
The absorption of water is facilitated when the polymer molecules have groups capable of forming hydrogen bonds. Vegetable fibers are rich in cellulose, hemicellulose, and lignin which have hydroxy groups, thus have high affinity for water. The absorption of water by the resin, in turn, can be considered practically null, since it presents a considerable hydrophobic character [38]. The addition of the vegetable fibers to the resin generates an increase in the water absorption levels, so an important parameter to be analyzed is how much water is being absorbed by the composite over time.
For the analysis of the free (C) and entrapped (S) water mass distributions inside the composite, the results of these parameters were plotted at different plans and times of process (x = R x /2, y = R y /2 and z = R z /2; t 1 6 h, t 2 54 h, t 3 211 h, and t 4 417 h). Figure 11 schematically illustrates the regions, in the Cartesian system, where the distributions of the parameters of interest will be analyzed. be considered practically null, since it presents a considerable hydrophobic character [38]. The addition of the vegetable fibers to the resin generates an increase in the water absorption levels, so an important parameter to be analyzed is how much water is being absorbed by the composite over time.
For the analysis of the free (C) and entrapped (S) water mass distributions inside the composite, the results of these parameters were plotted at different plans and times of process (x = Rx/2, y = Ry/2 and z = Rz/2; t1 ≅ 6 h, t2 ≅ 54 h, t3 ≅ 211 h, and t4 ≅ 417 h). Figure 11 schematically illustrates the regions, in the Cartesian system, where the distributions of the parameters of interest will be analyzed.           The variation of this parameter (l x ) demonstrates the amount of water in the x direction around the sample, in other words, changing l x means varying the position, in the x direction, of the immersed composite or the dimensions of the container used in the bath. Thus, with this analysis, it is possible to verify that these variations do not directly influence the absorption of moisture under the conditions studied. Thus, for the initial and boundary conditions established, the variation of the l x values does not lead to a significant change in the three-dimensional and transient moisture absorption parameters of the Langmuir model. Figures 15-17 illustrate the distributions of free solute concentration in the plans x = R x /2, z = R z /2, and y = R y /2, respectively, in a total time t = 211 h, for the cases where variation in the distance of the sample to the container wall (l x ) was stablished. values does not lead to a significant change in the three-dimensional and transient moisture absorption parameters of the Langmuir model. Figures 15-17 illustrate the distributions of free solute concentration in the plans x = Rx/2, z = Rz/2, and y = Ry/2, respectively, in a total time t = 211 h, for the cases where variation in the distance of the sample to the container wall (lx) was stablished. absorption parameters of the Langmuir model. Figures 15-17 illustrate the distributions of free solute concentration in the plans x = Rx/2, z = Rz/2, and y = Ry/2, respectively, in a total time t = 211 h, for the cases where variation in the distance of the sample to the container wall (lx) was stablished. From the analysis of these figures, it can be seen that in the plans x = Rx/2 and z = Rz/2, the increase in the value of lx does not affect the free solute concentration distribution, which confirms the independence of the behavior of the water absorption phenomenon studied, with this parameter lx. For the free solute concentration in the y = Ry/2 plan (Figure 16), in the case where lx = 0.01 m, there is a small reduction in free solute concentration, in the x-direction, compared with that in the z- From the analysis of these figures, it can be seen that in the plans x = Rx/2 and z = Rz/2, the increase in the value of lx does not affect the free solute concentration distribution, which confirms the independence of the behavior of the water absorption phenomenon studied, with this parameter lx. For the free solute concentration in the y = Ry/2 plan (Figure 16), in the case where lx = 0.01 m, there is a small reduction in free solute concentration, in the x-direction, compared with that in the zdirections.  From the analysis of these figures, it can be seen that in the plans x = R x /2 and z = R z /2, the increase in the value of l x does not affect the free solute concentration distribution, which confirms the independence of the behavior of the water absorption phenomenon studied, with this parameter l x . For the free solute concentration in the y = R y /2 plan (Figure 16), in the case where l x = 0.01 m, there is a small reduction in free solute concentration, in the x-direction, compared with that in the z-directions.
The distributions of the entrapped solute concentration in the plans x = R x /2, z = R z /2 and y = R y /2, at t = 211 h, keeping the values of l z and l y constant and varying the value of l x can be observed in  In this case, in neither plan is observed significant variations of the parameter S with the modification on the value of the sample distance to the container wall in the x direction (l x ). The distributions of the entrapped solute concentration in the plans x = Rx/2, z = Rz/2 and y = Ry/2, at t = 211 h, keeping the values of lz and ly constant and varying the value of lx can be observed in Figures 18-20. In this case, in neither plan is observed significant variations of the parameter S with the modification on the value of the sample distance to the container wall in the x direction (lx). The distributions of the entrapped solute concentration in the plans x = Rx/2, z = Rz/2 and y = Ry/2, at t = 211 h, keeping the values of lz and ly constant and varying the value of lx can be observed in Figures 18-20. In this case, in neither plan is observed significant variations of the parameter S with the modification on the value of the sample distance to the container wall in the x direction (lx). It is observed that the increase in the C and S concentrations of the Langmuir model occurs from the surface to the center of the composite. For the concentration of free solute (C), it is observed that the flux of moisture in the x and z direction occurs symmetrically since, in this case, the sample has the same dimensions (Rx = Rz). For the same time of process, the concentration of free solute showed higher values than the concentration of entrapped solute in the inner regions of the composite. This behavior is probably due to the effects of the adsorptive forces being smaller than those presented by the forces from the free water concentration gradient, which are responsible for the migration of water into the composite. For the concentration of entrapped solute, it is observed that the changes in this variable occur more slowly, however with the greatest intensity in the horizontal directions (x and z directions). It is observed that the increase in the C and S concentrations of the Langmuir model occurs from the surface to the center of the composite. For the concentration of free solute (C), it is observed that the flux of moisture in the x and z direction occurs symmetrically since, in this case, the sample has the same dimensions (R x = R z ). For the same time of process, the concentration of free solute showed higher values than the concentration of entrapped solute in the inner regions of the composite. This behavior is probably due to the effects of the adsorptive forces being smaller than those presented by the forces from the free water concentration gradient, which are responsible for the migration of water into the composite. For the concentration of entrapped solute, it is observed that the changes in this variable occur more slowly, however with the greatest intensity in the horizontal directions (x and z directions). From the analysis of the Figures 21-23, we can see that variations of l y values strongly influence the average free solute concentration, the average entrapped solute concentration, and average moisture content. The higher the l y value the higher the water layer close to the composite wall and faster shall be the water absorption rate. For initial times of the process, the differences among the predicted results are not significant, however in the course of time it can be clearly observed the influence of this parameter.         From the analysis of the Figures 21-23, we can see that variations of ly values strongly influence the average free solute concentration, the average entrapped solute concentration, and average moisture content. The higher the ly value the higher the water layer close to the composite wall and faster shall be the water absorption rate. For initial times of the process, the differences among the predicted results are not significant, however in the course of time it can be clearly observed the influence of this parameter. Figures 24-26 illustrate the free solute concentration distribution, in the plans x = Rx/2, z = Rz/2, and y = Ry/2, respectively, at t = 211 h, for different values of ly. After analysis of these figures, it is possible to observe that, in plans x = Rx/2 and z = Rz/2, for small value of ly (ly = 0.001 m), the flux of free solute practically does not occur in y direction, in opposite we can see it in the other directions. Then, it verified the free water molecules move horizontally (x and z direction), in practice, with velocity almost equal.   From the analysis of the Figures 21-23, we can see that variations of ly values strongly influence the average free solute concentration, the average entrapped solute concentration, and average moisture content. The higher the ly value the higher the water layer close to the composite wall and faster shall be the water absorption rate. For initial times of the process, the differences among the predicted results are not significant, however in the course of time it can be clearly observed the influence of this parameter. It can be observed a similar behavior that presented by free solute concentration. Then, in general, the geometric parameter l y affects distribution of the free and entrapped solute concentration and, in consequence, total moisture of the composite immersed in water. Since these are the first results on the subject that demonstrate that the parameter 1 y has influence on the kinetics of moisture absorption, the authors recommend strong l y new studies related to this topic, in order to classify this phenomenon. From the analysis of these figures, it is observed that the free solute concentration is going to increase from sample surface to center with a flux of moisture that is typically horizontal, that is, a greater flux occurs in the x and z direction comparing with that in the y-direction. The same behavior repeats when it is considered the entrapped solute concentration (Figures 27-29), nevertheless this concentration increases more slowly in comparison to the behavior of the free solute concentration, mainly in the initial times of process. It is observed no significant difference in the mass absorption phenomenon studied (moisture absorption) with the variation of lz. Then, it is possible to notice that only ly parameter can influence directly on the kinetics of moisture absorption in the studied conditions. However, for lz values lower than the 0.01 m, the effect of this geometrical parameter must appear more clearly.  From the analysis of these figures, it is observed that the free solute concentration is going to increase from sample surface to center with a flux of moisture that is typically horizontal, that is, a greater flux occurs in the x and z direction comparing with that in the y-direction. The same behavior repeats when it is considered the entrapped solute concentration (Figures 27-29), nevertheless this concentration increases more slowly in comparison to the behavior of the free solute concentration, mainly in the initial times of process. From the analysis of these figures, it is observed that the free solute concentration is going to increase from sample surface to center with a flux of moisture that is typically horizontal, that is, a greater flux occurs in the x and z direction comparing with that in the y-direction. The same behavior repeats when it is considered the entrapped solute concentration (Figures 27-29), nevertheless this concentration increases more slowly in comparison to the behavior of the free solute concentration, mainly in the initial times of process. It is observed no significant difference in the mass absorption phenomenon studied (moisture absorption) with the variation of lz. Then, it is possible to notice that only ly parameter can influence directly on the kinetics of moisture absorption in the studied conditions. However, for lz values lower than the 0.01 m, the effect of this geometrical parameter must appear more clearly.    It is noteworthy that the distributions of free and entrapped solute concentrations, for the different planes analyzed (x = Rx/ 2, y = Ry/ 2, z = Rz/ 2), at a given time and for different distances lz, occur similarly to the distributions of C and S found in the analysis of the influence of lx variation (Figures 15-20), in other words, there are no significant variations in these parameters changing the l distance in the z direction. Additionally, one must remember that, despite the probabilities of λ and µ to be equal, variation of entrapped solute concentration occurs more slowly comparing with the free solute concentration.

Effect of Distance lz
Thus, it can be said that only the ly parameter directly influences the moisture absorption kinetics under the studied conditions. The influence of this parameter is not contemplated by the Fick model, making the Langmuir model a more complete model since it can assimilate imperceptible effects in relation to the Fickian model (most used). Table 5 summarizes the cases treated for the analysis of the effect of distance lx, ly, and lz and their respective variations in the absorption parameters of the studied model. Table 5. Langmuir absorption parameter values found for t = 250 h and different lx, ly, and lz (arbitrary cases).  It is noteworthy that the distributions of free and entrapped solute concentrations, for the different planes analyzed (x = Rx/ 2, y = Ry/ 2, z = Rz/ 2), at a given time and for different distances lz, occur similarly to the distributions of C and S found in the analysis of the influence of lx variation (Figures 15-20), in other words, there are no significant variations in these parameters changing the l distance in the z direction. Additionally, one must remember that, despite the probabilities of λ and µ to be equal, variation of entrapped solute concentration occurs more slowly comparing with the free solute concentration.

Case t (h) Water Composite
Thus, it can be said that only the ly parameter directly influences the moisture absorption kinetics under the studied conditions. The influence of this parameter is not contemplated by the Fick model, making the Langmuir model a more complete model since it can assimilate imperceptible effects in relation to the Fickian model (most used). Table 5 summarizes the cases treated for the analysis of the effect of distance lx, ly, and lz and their respective variations in the absorption parameters of the studied model. Table 5. Langmuir absorption parameter values found for t = 250 h and different lx, ly, and lz (arbitrary cases). It is observed no significant difference in the mass absorption phenomenon studied (moisture absorption) with the variation of l z . Then, it is possible to notice that only l y parameter can influence directly on the kinetics of moisture absorption in the studied conditions. However, for l z values lower than the 0.01 m, the effect of this geometrical parameter must appear more clearly.

Case t (h) Water Composite
It is noteworthy that the distributions of free and entrapped solute concentrations, for the different planes analyzed (x = R x / 2, y = R y / 2, z = R z / 2), at a given time and for different distances l z , occur similarly to the distributions of C and S found in the analysis of the influence of lx variation (Figures 15-20), in other words, there are no significant variations in these parameters changing the l distance in the z direction. Additionally, one must remember that, despite the probabilities of λ and µ to be equal, variation of entrapped solute concentration occurs more slowly comparing with the free solute concentration.
Thus, it can be said that only the l y parameter directly influences the moisture absorption kinetics under the studied conditions. The influence of this parameter is not contemplated by the Fick model, making the Langmuir model a more complete model since it can assimilate imperceptible effects in relation to the Fickian model (most used). Table 5 summarizes the cases treated for the analysis of the effect of distance l x , l y , and l z and their respective variations in the absorption parameters of the studied model. Table 5. Langmuir absorption parameter values found for t = 250 h and different l x , l y , and l z (arbitrary cases).